Explore 3-1: Algebra Lab Analyzing Linear Graphs
1. Compare and contrast the key features of lines that
slant up and lines that slant down.
SOLUTION: Sample answer:
Domain and Range: For both lines that slant up and
lines that slant down, the domain and range are all real
numbers.
x- and y-intercepts: For both lines that slant up and lines that slant down, there is one x-intercept and one
y-intercept.
Extrema: For both lines that slant up and lines that
slant down, there are no maximum or minimum points.
Symmetry: Lines that slant up and lines that slant
down have no line symmetry.
Positive/Negative: For lines that slant up, the
function values are negative to the left of the xintercept and positive to the right. For lines that slant
down, the function values are positive to the left of the
x-intercept and negative to the right.
Increasing/Decreasing: For lines that slant up, the
function is increasing on the entire domain. For lines
that slant down, the function is decreasing on the
entire domain.
End Behavior: For lines that slant up, as x
decreases, y increases and as x increases, y
increases. For lines that slant down, as x decreases, y
increases and as x increases, y decreases.
negative instead of all positive.
Increasing/Decreasing: The function is constant on
the entire domain.
End Behavior: The end behavior is the same.
3. Consider lines that pass through the origin.
a. How do the key features of a line that slants up and
passes through the origin compare to the key features
of the line in Activity 1?
b. Compare the key features of a line that slants down
and passes through the origin to the key features of
the line in Activity 2.
c. Describe a horizontal line that passes through the
origin and a vertical line that passes through the origin.
Compare their key features to those of the lines in
Activities 3 and 4.
SOLUTION: a. The key features are the same except that the xand y-intercepts are the same point, (0, 0).
b. The key features are the same except that the xand y-intercepts are the same point, (0, 0).
2. How would the key features of a horizontal line below
the x-axis differ from the features of a line above the
x-axis?
SOLUTION: Sample answer:
Domain/Range: The domain is still all real numbers.
The range is one negative value instead of a positive
value.
x- and y-intercepts: There is still no x-intercept.
The one y-intercept is a negative value instead of a
positive value.
Extrema: There are still no maximum or minimum
points.
Symmetry: The graph is still symmetric about any
vertical line.
Positive/Negative: The function values are all
negative instead of all positive.
Increasing/Decreasing: The function is constant on
the entire domain.
End Behavior: The end behavior is the same.
3. Consider lines that pass through the origin.
a. How do the key features of a line that slants up and
eSolutions
- Powered
by Cognero
passesManual
through
the origin
compare to the key features
of the line in Activity 1?
b. Compare the key features of a line that slants down
c. The horizontal line that passes through the origin is
the x-axis. The key features of the graph are the same
as those of the line in Activity 3 except that instead of
no x-intercept, it has infinitely many x-intercepts. The
vertical line that passes through the origin is the y-axis.
The key features of the graph are the same as those
of the line in Activity 4 except that instead of no yintercept, it has infinitely many y-intercepts.
Page 1
as those of the line in Activity 3 except that instead of
no x-intercept, it has infinitely many x-intercepts. The
vertical line that passes through the origin is the y-axis.
The key
features
of Lab
the graph
are theLinear
same as
those
Explore
3-1:
Algebra
Analyzing
Graphs
of the line in Activity 4 except that instead of no yintercept, it has infinitely many y-intercepts.
left to right. Because a line is straight, it will slope
upward over the entire domain. This indicates that the
function is increasing over the entire domain.
c. False; no two unique lines can have the same xand y-intercepts.
Sketch a linear graph that fits each description.
5. as x increases, y decreases
SOLUTION: As x increases, y decreases for any line that slants
downward from left to right.
Sample graph: 4. CCSS TOOLS Place a pencil on a coordinate plane to represent a line. Move the pencil to represent
different lines and evaluate each conjecture.
a. True or false: A line can have more than one xintercept. b. True or false: If the end behavior of a line is that as
x increases, y increases, then the function values are
increasing over the entire domain. c. True or false: Two different lines can have the
same x- and y-intercepts.
SOLUTION: a. True; the x-axis is the one line that has more than
one x-intercept. It has infinitely many x-intercepts.
Other horizontal lines have no x-intercept, and all other
lines have exactly one.
b. True; a line with end behavior slopes upward from
left to right. Because a line is straight, it will slope
upward over the entire domain. This indicates that the
function is increasing over the entire domain.
c. False; no two unique lines can have the same xand y-intercepts.
Sketch a linear graph that fits each description.
5. as x increases, y decreases
6. one x-intercept and one y-intercept
SOLUTION: Any nonvertical, nonhorizontal line has one xintercept and one y -intercept.
Sample graph:
7. has symmetry
SOLUTION: Any vertical or horizontal line has line symmetry.
Sample graph:
SOLUTION: As x increases, y decreases for any line that slants
downward from left to right.
Sample graph: 8. is not a function
SOLUTION: eSolutions Manual - Powered by Cognero
6. one x-intercept and one y-intercept
Any vertical line is not a function.
Sample graph:
Page 2
Explore 3-1: Algebra Lab Analyzing Linear Graphs
8. is not a function
SOLUTION: Any vertical line is not a function.
Sample graph:
eSolutions Manual - Powered by Cognero
Page 3